# [FOM] Characterization of the real numbers

Mon Feb 7 15:30:11 EST 2005

```X (a complete ordered set) having a countable dense subset means that
there is a function F: X -> X and a z in X such that the orbit of F
starting at z (closure of {z} under F) is dense in X.

Best,  Bill Tait

On Feb 6, 2005, at 7:10 PM, JoeShipman at aol.com wrote:

> Yes, but "having a countable dense subset" involves significantly
> higher logical complexity, you have to define "countable subset".  I'd
> like to avoid defining the integers if possible.
>
> Mayberry wrote:
> How about "complete linear ordering without endpoints having a
> countable
> dense subset". I think this is Cantor's characterization.
>
> --On 06 February 2005 15:23 -0500 JoeShipman at aol.com wrote:
>
>> What is the simplest characterization of the real numbers?  That is,
>> what
>> is the simplest description of a structure, any model of which is
>> isomorphic to the real numbers?
>>
>> A standard way of characterizing the real numbers is "ordered field
>> with
>> the least upper bound property".  But do I need to refer to field
>> operations?  "Dense ordering without endpoints and the least upper
>> bound
>> property" isn't sharp enough.  "Homogenous dense ordering with the
>> least
>> upper bound property" looks better, except that it doesn't rule out
>> the
>> "long line" (product of the set of countable ordinals with [0,1} in
>> dictionary order, with intial point removed).  (It also doesn't rule
>> out
>> the reversed long line, or the symmetric long line.)
>>
>> The best I can do without referring to relations other than the order
>> relation is "dense ordering with least upper bound property,
>> isomorphic
>> to any of its nonempty open intervals". Can anyone improve on this?
>>
>> -- JS
>
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